3.4.84 \(\int x^3 \sinh (a+b x) \tanh ^2(a+b x) \, dx\) [384]

Optimal. Leaf size=162 \[ -\frac {6 x^2 \text {ArcTan}\left (e^{a+b x}\right )}{b^2}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}+\frac {6 i x \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}-\frac {6 i x \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}-\frac {6 i \text {PolyLog}\left (3,-i e^{a+b x}\right )}{b^4}+\frac {6 i \text {PolyLog}\left (3,i e^{a+b x}\right )}{b^4}+\frac {x^3 \text {sech}(a+b x)}{b}-\frac {6 \sinh (a+b x)}{b^4}-\frac {3 x^2 \sinh (a+b x)}{b^2} \]

[Out]

-6*x^2*arctan(exp(b*x+a))/b^2+6*x*cosh(b*x+a)/b^3+x^3*cosh(b*x+a)/b+6*I*x*polylog(2,-I*exp(b*x+a))/b^3-6*I*x*p
olylog(2,I*exp(b*x+a))/b^3-6*I*polylog(3,-I*exp(b*x+a))/b^4+6*I*polylog(3,I*exp(b*x+a))/b^4+x^3*sech(b*x+a)/b-
6*sinh(b*x+a)/b^4-3*x^2*sinh(b*x+a)/b^2

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Rubi [A]
time = 0.15, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5557, 3377, 2717, 5526, 4265, 2611, 2320, 6724} \begin {gather*} -\frac {6 x^2 \text {ArcTan}\left (e^{a+b x}\right )}{b^2}-\frac {6 i \text {Li}_3\left (-i e^{a+b x}\right )}{b^4}+\frac {6 i \text {Li}_3\left (i e^{a+b x}\right )}{b^4}-\frac {6 \sinh (a+b x)}{b^4}+\frac {6 i x \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {6 i x \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {6 x \cosh (a+b x)}{b^3}-\frac {3 x^2 \sinh (a+b x)}{b^2}+\frac {x^3 \cosh (a+b x)}{b}+\frac {x^3 \text {sech}(a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Sinh[a + b*x]*Tanh[a + b*x]^2,x]

[Out]

(-6*x^2*ArcTan[E^(a + b*x)])/b^2 + (6*x*Cosh[a + b*x])/b^3 + (x^3*Cosh[a + b*x])/b + ((6*I)*x*PolyLog[2, (-I)*
E^(a + b*x)])/b^3 - ((6*I)*x*PolyLog[2, I*E^(a + b*x)])/b^3 - ((6*I)*PolyLog[3, (-I)*E^(a + b*x)])/b^4 + ((6*I
)*PolyLog[3, I*E^(a + b*x)])/b^4 + (x^3*Sech[a + b*x])/b - (6*Sinh[a + b*x])/b^4 - (3*x^2*Sinh[a + b*x])/b^2

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5526

Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*Tanh[(a_.) + (b_.)*(x_)^(n_.)]^(q_.), x_Symbol] :> Simp[(-
x^(m - n + 1))*(Sech[a + b*x^n]^p/(b*n*p)), x] + Dist[(m - n + 1)/(b*n*p), Int[x^(m - n)*Sech[a + b*x^n]^p, x]
, x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]

Rule 5557

Int[((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int
[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int x^3 \sinh (a+b x) \tanh ^2(a+b x) \, dx &=\int x^3 \sinh (a+b x) \, dx-\int x^3 \text {sech}(a+b x) \tanh (a+b x) \, dx\\ &=\frac {x^3 \cosh (a+b x)}{b}+\frac {x^3 \text {sech}(a+b x)}{b}-\frac {3 \int x^2 \cosh (a+b x) \, dx}{b}-\frac {3 \int x^2 \text {sech}(a+b x) \, dx}{b}\\ &=-\frac {6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {x^3 \cosh (a+b x)}{b}+\frac {x^3 \text {sech}(a+b x)}{b}-\frac {3 x^2 \sinh (a+b x)}{b^2}+\frac {(6 i) \int x \log \left (1-i e^{a+b x}\right ) \, dx}{b^2}-\frac {(6 i) \int x \log \left (1+i e^{a+b x}\right ) \, dx}{b^2}+\frac {6 \int x \sinh (a+b x) \, dx}{b^2}\\ &=-\frac {6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}+\frac {6 i x \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {6 i x \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {x^3 \text {sech}(a+b x)}{b}-\frac {3 x^2 \sinh (a+b x)}{b^2}-\frac {(6 i) \int \text {Li}_2\left (-i e^{a+b x}\right ) \, dx}{b^3}+\frac {(6 i) \int \text {Li}_2\left (i e^{a+b x}\right ) \, dx}{b^3}-\frac {6 \int \cosh (a+b x) \, dx}{b^3}\\ &=-\frac {6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}+\frac {6 i x \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {6 i x \text {Li}_2\left (i e^{a+b x}\right )}{b^3}+\frac {x^3 \text {sech}(a+b x)}{b}-\frac {6 \sinh (a+b x)}{b^4}-\frac {3 x^2 \sinh (a+b x)}{b^2}-\frac {(6 i) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {(6 i) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac {6 x^2 \tan ^{-1}\left (e^{a+b x}\right )}{b^2}+\frac {6 x \cosh (a+b x)}{b^3}+\frac {x^3 \cosh (a+b x)}{b}+\frac {6 i x \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}-\frac {6 i x \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {6 i \text {Li}_3\left (-i e^{a+b x}\right )}{b^4}+\frac {6 i \text {Li}_3\left (i e^{a+b x}\right )}{b^4}+\frac {x^3 \text {sech}(a+b x)}{b}-\frac {6 \sinh (a+b x)}{b^4}-\frac {3 x^2 \sinh (a+b x)}{b^2}\\ \end {align*}

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Mathematica [A]
time = 2.28, size = 196, normalized size = 1.21 \begin {gather*} \frac {-3 i \left (b^2 x^2 \log \left (1-i e^{a+b x}\right )-b^2 x^2 \log \left (1+i e^{a+b x}\right )-2 b x \text {PolyLog}\left (2,-i e^{a+b x}\right )+2 b x \text {PolyLog}\left (2,i e^{a+b x}\right )+2 \text {PolyLog}\left (3,-i e^{a+b x}\right )-2 \text {PolyLog}\left (3,i e^{a+b x}\right )\right )+b^3 x^3 \text {sech}(a+b x)+\cosh (b x) \left (b x \left (6+b^2 x^2\right ) \cosh (a)-3 \left (2+b^2 x^2\right ) \sinh (a)\right )+\left (-3 \left (2+b^2 x^2\right ) \cosh (a)+b x \left (6+b^2 x^2\right ) \sinh (a)\right ) \sinh (b x)}{b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sinh[a + b*x]*Tanh[a + b*x]^2,x]

[Out]

((-3*I)*(b^2*x^2*Log[1 - I*E^(a + b*x)] - b^2*x^2*Log[1 + I*E^(a + b*x)] - 2*b*x*PolyLog[2, (-I)*E^(a + b*x)]
+ 2*b*x*PolyLog[2, I*E^(a + b*x)] + 2*PolyLog[3, (-I)*E^(a + b*x)] - 2*PolyLog[3, I*E^(a + b*x)]) + b^3*x^3*Se
ch[a + b*x] + Cosh[b*x]*(b*x*(6 + b^2*x^2)*Cosh[a] - 3*(2 + b^2*x^2)*Sinh[a]) + (-3*(2 + b^2*x^2)*Cosh[a] + b*
x*(6 + b^2*x^2)*Sinh[a])*Sinh[b*x])/b^4

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Maple [F]
time = 1.60, size = 0, normalized size = 0.00 \[\int x^{3} \mathrm {sech}\left (b x +a \right )^{2} \left (\sinh ^{3}\left (b x +a \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*sech(b*x+a)^2*sinh(b*x+a)^3,x)

[Out]

int(x^3*sech(b*x+a)^2*sinh(b*x+a)^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sech(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

1/2*((b^3*x^3*e^(4*a) - 3*b^2*x^2*e^(4*a) + 6*b*x*e^(4*a) - 6*e^(4*a))*e^(3*b*x) + 6*(b^3*x^3*e^(2*a) + 2*b*x*
e^(2*a))*e^(b*x) + (b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^(-b*x))/(b^4*e^(2*b*x + 3*a) + b^4*e^a) - 6*integrate(x
^2*e^(b*x + a)/(b*e^(2*b*x + 2*a) + b), x)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1225 vs. \(2 (139) = 278\).
time = 0.41, size = 1225, normalized size = 7.56 \begin {gather*} \frac {b^{3} x^{3} + {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \sinh \left (b x + a\right )^{4} + 3 \, b^{2} x^{2} + 6 \, {\left (b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{2} + 6 \, {\left (b^{3} x^{3} + {\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right )^{2} + 2 \, b x\right )} \sinh \left (b x + a\right )^{2} + 6 \, b x - 12 \, {\left (i \, b x \cosh \left (b x + a\right )^{3} + 3 i \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + i \, b x \sinh \left (b x + a\right )^{3} + i \, b x \cosh \left (b x + a\right ) + {\left (3 i \, b x \cosh \left (b x + a\right )^{2} + i \, b x\right )} \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 12 \, {\left (-i \, b x \cosh \left (b x + a\right )^{3} - 3 i \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - i \, b x \sinh \left (b x + a\right )^{3} - i \, b x \cosh \left (b x + a\right ) + {\left (-3 i \, b x \cosh \left (b x + a\right )^{2} - i \, b x\right )} \sinh \left (b x + a\right )\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) - 6 \, {\left (i \, a^{2} \cosh \left (b x + a\right )^{3} + 3 i \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + i \, a^{2} \sinh \left (b x + a\right )^{3} + i \, a^{2} \cosh \left (b x + a\right ) + {\left (3 i \, a^{2} \cosh \left (b x + a\right )^{2} + i \, a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - 6 \, {\left (-i \, a^{2} \cosh \left (b x + a\right )^{3} - 3 i \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - i \, a^{2} \sinh \left (b x + a\right )^{3} - i \, a^{2} \cosh \left (b x + a\right ) + {\left (-3 i \, a^{2} \cosh \left (b x + a\right )^{2} - i \, a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - 6 \, {\left ({\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \sinh \left (b x + a\right )^{3} + {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \cosh \left (b x + a\right ) + {\left (-i \, b^{2} x^{2} + 3 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \cosh \left (b x + a\right )^{2} + i \, a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - 6 \, {\left ({\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \sinh \left (b x + a\right )^{3} + {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \cosh \left (b x + a\right ) + {\left (i \, b^{2} x^{2} + 3 \, {\left (i \, b^{2} x^{2} - i \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - i \, a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) - 12 \, {\left (-i \, \cosh \left (b x + a\right )^{3} - 3 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - i \, \sinh \left (b x + a\right )^{3} + {\left (-3 i \, \cosh \left (b x + a\right )^{2} - i\right )} \sinh \left (b x + a\right ) - i \, \cosh \left (b x + a\right )\right )} {\rm polylog}\left (3, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 12 \, {\left (i \, \cosh \left (b x + a\right )^{3} + 3 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + i \, \sinh \left (b x + a\right )^{3} + {\left (3 i \, \cosh \left (b x + a\right )^{2} + i\right )} \sinh \left (b x + a\right ) + i \, \cosh \left (b x + a\right )\right )} {\rm polylog}\left (3, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 4 \, {\left ({\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 6}{2 \, {\left (b^{4} \cosh \left (b x + a\right )^{3} + 3 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{4} \sinh \left (b x + a\right )^{3} + b^{4} \cosh \left (b x + a\right ) + {\left (3 \, b^{4} \cosh \left (b x + a\right )^{2} + b^{4}\right )} \sinh \left (b x + a\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sech(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/2*(b^3*x^3 + (b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a)^4 + 4*(b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*
x + a)*sinh(b*x + a)^3 + (b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*sinh(b*x + a)^4 + 3*b^2*x^2 + 6*(b^3*x^3 + 2*b*x)*c
osh(b*x + a)^2 + 6*(b^3*x^3 + (b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a)^2 + 2*b*x)*sinh(b*x + a)^2 + 6*b
*x - 12*(I*b*x*cosh(b*x + a)^3 + 3*I*b*x*cosh(b*x + a)*sinh(b*x + a)^2 + I*b*x*sinh(b*x + a)^3 + I*b*x*cosh(b*
x + a) + (3*I*b*x*cosh(b*x + a)^2 + I*b*x)*sinh(b*x + a))*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 12*(-I*b*
x*cosh(b*x + a)^3 - 3*I*b*x*cosh(b*x + a)*sinh(b*x + a)^2 - I*b*x*sinh(b*x + a)^3 - I*b*x*cosh(b*x + a) + (-3*
I*b*x*cosh(b*x + a)^2 - I*b*x)*sinh(b*x + a))*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) - 6*(I*a^2*cosh(b*x +
a)^3 + 3*I*a^2*cosh(b*x + a)*sinh(b*x + a)^2 + I*a^2*sinh(b*x + a)^3 + I*a^2*cosh(b*x + a) + (3*I*a^2*cosh(b*x
+ a)^2 + I*a^2)*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + I) - 6*(-I*a^2*cosh(b*x + a)^3 - 3*I*a^2*c
osh(b*x + a)*sinh(b*x + a)^2 - I*a^2*sinh(b*x + a)^3 - I*a^2*cosh(b*x + a) + (-3*I*a^2*cosh(b*x + a)^2 - I*a^2
)*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) - I) - 6*((-I*b^2*x^2 + I*a^2)*cosh(b*x + a)^3 + 3*(-I*b^2*
x^2 + I*a^2)*cosh(b*x + a)*sinh(b*x + a)^2 + (-I*b^2*x^2 + I*a^2)*sinh(b*x + a)^3 + (-I*b^2*x^2 + I*a^2)*cosh(
b*x + a) + (-I*b^2*x^2 + 3*(-I*b^2*x^2 + I*a^2)*cosh(b*x + a)^2 + I*a^2)*sinh(b*x + a))*log(I*cosh(b*x + a) +
I*sinh(b*x + a) + 1) - 6*((I*b^2*x^2 - I*a^2)*cosh(b*x + a)^3 + 3*(I*b^2*x^2 - I*a^2)*cosh(b*x + a)*sinh(b*x +
a)^2 + (I*b^2*x^2 - I*a^2)*sinh(b*x + a)^3 + (I*b^2*x^2 - I*a^2)*cosh(b*x + a) + (I*b^2*x^2 + 3*(I*b^2*x^2 -
I*a^2)*cosh(b*x + a)^2 - I*a^2)*sinh(b*x + a))*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) - 12*(-I*cosh(b*x +
a)^3 - 3*I*cosh(b*x + a)*sinh(b*x + a)^2 - I*sinh(b*x + a)^3 + (-3*I*cosh(b*x + a)^2 - I)*sinh(b*x + a) - I*c
osh(b*x + a))*polylog(3, I*cosh(b*x + a) + I*sinh(b*x + a)) - 12*(I*cosh(b*x + a)^3 + 3*I*cosh(b*x + a)*sinh(b
*x + a)^2 + I*sinh(b*x + a)^3 + (3*I*cosh(b*x + a)^2 + I)*sinh(b*x + a) + I*cosh(b*x + a))*polylog(3, -I*cosh(
b*x + a) - I*sinh(b*x + a)) + 4*((b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*cosh(b*x + a)^3 + 3*(b^3*x^3 + 2*b*x)*cosh(
b*x + a))*sinh(b*x + a) + 6)/(b^4*cosh(b*x + a)^3 + 3*b^4*cosh(b*x + a)*sinh(b*x + a)^2 + b^4*sinh(b*x + a)^3
+ b^4*cosh(b*x + a) + (3*b^4*cosh(b*x + a)^2 + b^4)*sinh(b*x + a))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*sech(b*x+a)**2*sinh(b*x+a)**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*sech(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x^3*sech(b*x + a)^2*sinh(b*x + a)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*sinh(a + b*x)^3)/cosh(a + b*x)^2,x)

[Out]

int((x^3*sinh(a + b*x)^3)/cosh(a + b*x)^2, x)

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